Answer to Question #290341 in Discrete Mathematics for isara

Question

Let A = {a,b,c,d}, B = {1,2,3}, and R = {(a,2), (b, 1), (c, 2), (d, 1)}.

(a)Is R a function?

(b)Is R−1 a function?

Explain your answers.

Answer 1

(a) Since there are no pairs with the same first component, the relation $R$ is a function.

(b) It follows that $R^{-1}=\{(2,a),(1,b),(2,c),(1,d)\}.$

Since there are two pairs $(1,b)$ and $(1,d)$ with the same first component, we conclude that $R^{-1}$ is not a function.